PageRank, a Look at Small Changes in a Line of Nodes and the Complete Graph

  • Christopher EngströmEmail author
  • Sergei Silvestrov
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 179)


In this article we will look at the PageRank algorithm used as part of the ranking process of different Internet pages in search engines by for example Google. This article has its main focus in the understanding of the behavior of PageRank as the system dynamically changes either by contracting or expanding such as when adding or subtracting nodes or links or groups of nodes or links. In particular we will take a look at link structures consisting of a line of nodes or a complete graph where every node links to all others. We will look at PageRank as the solution of a linear system of equations and do our examination in both the ordinary normalized version of PageRank as well as the non-normalized version found by solving corresponding linear system. We will show that using two different methods we can find explicit formulas for the PageRank of some simple link structures.


PageRank Graph Random walk Block matrix 



This research was supported in part by the Swedish Research Council (621- 2007-6338), Swedish Foundation for International Cooperation in Research and Higher Education (STINT), Royal Swedish Academy of Sciences, Royal Physiographic Society in Lund and Crafoord Foundation.


  1. 1.
    Andersson, F.: Estimation of the quality of hyperlinked documents using a series formulation of pagerank. Master’s thesis, Mathematics, Centre for Mathematical sciences, Lund Institute of Technology, Lund University (2006:E22). LUTFMA-3132-2006Google Scholar
  2. 2.
    Andersson, F., Silvestrov, S.: The mathematics of internet search engines. Acta Appl. Math. 104, 211–242 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Berman, A., Plemmons, R.: Nonnegative Matrices in the Mathematical Sciences. No. Del 11 in Classics in Applied Mathematics. Society for Industrial and Applied Mathematics. Academic Press, New York (1994)Google Scholar
  4. 4.
    Bernstein, D.: Matrix Mathematics. Princeton University Press, Princeton (2005)Google Scholar
  5. 5.
    Bianchini, M., Gori, M., Scarselli, F.: Inside pagerank. ACM Trans. Int. Technol. 5(1), 92–128 (2005)CrossRefGoogle Scholar
  6. 6.
    Brin, S., Page, L.: The anatomy of a large-scale hypertextual web search engine. Comput. Netw. ISDN Syst. 30(1–7), 107–117 (1998) (Proceedings of the Seventh International World Wide Web Conference)Google Scholar
  7. 7.
    Bryan, K., Leise, T.: The $ 25,000,000,000 eigenvector: the linear algebra behind google. SIAM Rev. 48(3), 569–581 (2006)Google Scholar
  8. 8.
    Dhyani, D., Bhowmick, S.S., Ng, W.K.: Deriving and verifying statistical distribution of a hyperlink-based web page quality metric. Data Knowl. Eng. 46(3), 291–315 (2003)CrossRefzbMATHGoogle Scholar
  9. 9.
    Engström, C., Silvestrov, S.: Non-normalized pagerank and random walks on n-partite graphs. In: Proceedings of 3rd Stochastic Modeling Techniques and Data Analysis, pp. 192–202 (2015)Google Scholar
  10. 10.
    Gantmacher, F.: The Theory of Matrices. Chelsea, New York (1959)zbMATHGoogle Scholar
  11. 11.
    Haveliwala, T., Kamvar, S.: The second eigenvalue of the google matrix. Technical Report 2003–20, Stanford InfoLab (2003)Google Scholar
  12. 12.
    Ishii, H., Tempo, R., Bai, E.W., Dabbene, F.: Distributed randomized pagerank computation based on web aggregation. In: Proceedings of the 48th IEEE Conference on Decision and Control, 2009 Held Jointly with the 2009 28th Chinese Control Conference. CDC/CCC 2009, pp. 3026–3031 (2009)Google Scholar
  13. 13.
    Kamvar, S., Haveliwala, T.: The condition number of the pagerank problem. Technical Report 2003–36, Stanford InfoLab (2003)Google Scholar
  14. 14.
    Kamvar, S.D., Schlosser, M.T., Garcia-Molina, H.: The eigentrust algorithm for reputation management in P2P networks. In: Proceedings of the 12th International Conference on World Wide Web. WWW ’03, pp. 640–651. ACM, New York (2003)Google Scholar
  15. 15.
    Lancaster, P.: Theory of Matrices. Academic Press, New York (1969)Google Scholar
  16. 16.
    Norris, J.R.: Markov Chains. Cambridge University Press, New York (2009)zbMATHGoogle Scholar
  17. 17.
    Rydén, T., Lindgren, G.: Markovprocesser. Lund University, Lund (2000)Google Scholar
  18. 18.
    Sepandar, K., Taher, H., Gene, G.: Adaptive methods for the computation of pagerank. Linear Algebra Appl. 386(0), 51–65 (2004) (Special Issue on the Conference on the Numerical Solution of Markov Chains 2003)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Division of Applied Mathematics, School of EducationCulture and Communication, Mälardalen UniversityVästeråsSweden

Personalised recommendations